I'm looking on rigzone at HSE positions and I see nothing that wants less than 3 years of work experience. Can I apply for these with less than that with reasonable expectations, or is it usually pretty set in stone and I'd be wasting my time?

If you didn't graduate in HSE or really know somebody you better get some field experience in whatever field you want to enter. Whether it be petrochemical or offshore or whatever it's nice to actually see and know what hazards exist before you get paid to identify and mitigate them.